Integrand size = 16, antiderivative size = 264 \[ \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx=-\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}-\frac {5 a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}+\frac {5 a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}-\frac {5 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}}+\frac {5 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}} \]
-5/32*a*x*(-b*x^4+a)^(3/4)/b^2-1/8*x^5*(-b*x^4+a)^(3/4)/b+5/128*a^2*arctan (-1+b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4))/b^(9/4)*2^(1/2)+5/128*a^2*arctan(1 +b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4))/b^(9/4)*2^(1/2)-5/256*a^2*ln(1-b^(1/4 )*x*2^(1/2)/(-b*x^4+a)^(1/4)+x^2*b^(1/2)/(-b*x^4+a)^(1/2))/b^(9/4)*2^(1/2) +5/256*a^2*ln(1+b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4)+x^2*b^(1/2)/(-b*x^4+a)^ (1/2))/b^(9/4)*2^(1/2)
Time = 0.85 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.60 \[ \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx=\frac {-4 \sqrt [4]{b} x \left (a-b x^4\right )^{3/4} \left (5 a+4 b x^4\right )-5 \sqrt {2} a^2 \arctan \left (\frac {-\sqrt {b} x^2+\sqrt {a-b x^4}}{\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}\right )+5 \sqrt {2} a^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}{\sqrt {b} x^2+\sqrt {a-b x^4}}\right )}{128 b^{9/4}} \]
(-4*b^(1/4)*x*(a - b*x^4)^(3/4)*(5*a + 4*b*x^4) - 5*Sqrt[2]*a^2*ArcTan[(-( Sqrt[b]*x^2) + Sqrt[a - b*x^4])/(Sqrt[2]*b^(1/4)*x*(a - b*x^4)^(1/4))] + 5 *Sqrt[2]*a^2*ArcTanh[(Sqrt[2]*b^(1/4)*x*(a - b*x^4)^(1/4))/(Sqrt[b]*x^2 + Sqrt[a - b*x^4])])/(128*b^(9/4))
Time = 0.41 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {843, 843, 770, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {5 a \int \frac {x^4}{\sqrt [4]{a-b x^4}}dx}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {5 a \left (\frac {a \int \frac {1}{\sqrt [4]{a-b x^4}}dx}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {5 a \left (\frac {a \int \frac {1}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {5 a \left (\frac {a \left (\frac {1}{2} \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}+\frac {1}{2} \int \frac {\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}\right )}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5 a \left (\frac {a \left (\frac {1}{2} \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}+\frac {1}{2} \left (\frac {\int \frac {1}{\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )\right )}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 a \left (\frac {a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\frac {x^2}{\sqrt {a-b x^4}}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-\frac {x^2}{\sqrt {a-b x^4}}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}\right )}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 a \left (\frac {a \left (\frac {1}{2} \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5 a \left (\frac {a \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 a \left (\frac {a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 a \left (\frac {a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt {b}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1}{\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 a \left (\frac {a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )\right )}{4 b}-\frac {x \left (a-b x^4\right )^{3/4}}{4 b}\right )}{8 b}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}\) |
-1/8*(x^5*(a - b*x^4)^(3/4))/b + (5*a*(-1/4*(x*(a - b*x^4)^(3/4))/b + (a*( (-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(Sqrt[2]*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(Sqrt[2]*b^(1/4)))/2 + ( -1/2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^ 4)^(1/4)]/(Sqrt[2]*b^(1/4)) + Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqr t[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/4)))/2))/(4*b)))/(8*b)
3.13.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 4.78 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.76
| method | result | size |
| pseudoelliptic | \(\frac {-32 b^{\frac {5}{4}} x^{5} \left (-b \,x^{4}+a \right )^{\frac {3}{4}}-40 a x \,b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {3}{4}}-5 \sqrt {2}\, \ln \left (\frac {-b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2} \sqrt {b}+\sqrt {-b \,x^{4}+a}}{b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2} \sqrt {b}+\sqrt {-b \,x^{4}+a}}\right ) a^{2}-10 \arctan \left (\frac {b^{\frac {1}{4}} x +\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) \sqrt {2}\, a^{2}+10 \arctan \left (\frac {b^{\frac {1}{4}} x -\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) \sqrt {2}\, a^{2}}{256 b^{\frac {9}{4}}}\) | \(200\) |
1/256*(-32*b^(5/4)*x^5*(-b*x^4+a)^(3/4)-40*a*x*b^(1/4)*(-b*x^4+a)^(3/4)-5* 2^(1/2)*ln((-b^(1/4)*(-b*x^4+a)^(1/4)*2^(1/2)*x+x^2*b^(1/2)+(-b*x^4+a)^(1/ 2))/(b^(1/4)*(-b*x^4+a)^(1/4)*2^(1/2)*x+x^2*b^(1/2)+(-b*x^4+a)^(1/2)))*a^2 -10*arctan((b^(1/4)*x+2^(1/2)*(-b*x^4+a)^(1/4))/b^(1/4)/x)*2^(1/2)*a^2+10* arctan((b^(1/4)*x-2^(1/2)*(-b*x^4+a)^(1/4))/b^(1/4)/x)*2^(1/2)*a^2)/b^(9/4 )
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.90 \[ \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx=-\frac {5 \, b^{2} \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {125 \, {\left (b^{7} x \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {3}{4}} + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{6}\right )}}{x}\right ) - 5 \, b^{2} \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {125 \, {\left (b^{7} x \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {3}{4}} - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{6}\right )}}{x}\right ) + 5 i \, b^{2} \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {125 \, {\left (i \, b^{7} x \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {3}{4}} - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{6}\right )}}{x}\right ) - 5 i \, b^{2} \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {125 \, {\left (-i \, b^{7} x \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {3}{4}} - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{6}\right )}}{x}\right ) + 4 \, {\left (4 \, b x^{5} + 5 \, a x\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{128 \, b^{2}} \]
-1/128*(5*b^2*(-a^8/b^9)^(1/4)*log(125*(b^7*x*(-a^8/b^9)^(3/4) + (-b*x^4 + a)^(1/4)*a^6)/x) - 5*b^2*(-a^8/b^9)^(1/4)*log(-125*(b^7*x*(-a^8/b^9)^(3/4 ) - (-b*x^4 + a)^(1/4)*a^6)/x) + 5*I*b^2*(-a^8/b^9)^(1/4)*log(-125*(I*b^7* x*(-a^8/b^9)^(3/4) - (-b*x^4 + a)^(1/4)*a^6)/x) - 5*I*b^2*(-a^8/b^9)^(1/4) *log(-125*(-I*b^7*x*(-a^8/b^9)^(3/4) - (-b*x^4 + a)^(1/4)*a^6)/x) + 4*(4*b *x^5 + 5*a*x)*(-b*x^4 + a)^(3/4))/b^2
Result contains complex when optimal does not.
Time = 1.55 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.15 \[ \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx=\frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac {13}{4}\right )} \]
x**9*gamma(9/4)*hyper((1/4, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4* a**(1/4)*gamma(13/4))
Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00 \[ \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx=-\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {b} + \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} - \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {1}{4}}}\right )} a^{2}}{256 \, b^{2}} - \frac {\frac {9 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} a^{2} b}{x^{3}} + \frac {5 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} a^{2}}{x^{7}}}{32 \, {\left (b^{4} - \frac {2 \, {\left (b x^{4} - a\right )} b^{3}}{x^{4}} + \frac {{\left (b x^{4} - a\right )}^{2} b^{2}}{x^{8}}\right )}} \]
-5/256*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(-b*x^4 + a)^(1/ 4)/x)/b^(1/4))/b^(1/4) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4))/b^(1/4) - sqrt(2)*log(sqrt(b) + sqrt(2)*( -b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(1/4) + sqrt(2)*log( sqrt(b) - sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^( 1/4))*a^2/b^2 - 1/32*(9*(-b*x^4 + a)^(3/4)*a^2*b/x^3 + 5*(-b*x^4 + a)^(7/4 )*a^2/x^7)/(b^4 - 2*(b*x^4 - a)*b^3/x^4 + (b*x^4 - a)^2*b^2/x^8)
\[ \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx=\int { \frac {x^{8}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx=\int \frac {x^8}{{\left (a-b\,x^4\right )}^{1/4}} \,d x \]